Color groups arising from index-nsubgroups of symmetry groups

Abstract

One of the main goals in the study of color symmetry is to classify colorings of symmetrical objects through their color groups. The term color group is taken to mean the subgroup of the symmetry group of the uncolored symmetrical object which induces a permutation of colors in the coloring. This work looks for methods of determining the color group of a colored symmetric object. It begins with an index n subgroup H of the symmetry group G of the uncolored object. It then considers H-invariant colorings of the object, so that the color group H(*) will be a subgroup of G containing H. In other words, H ≤ H(*) ≤ G. It proceeds to give necessary and sufficient conditions for the equality of H(*) and G. If H(*) ≠ G and n is prime, then H(*) = H. On the other hand, if H(*) ≠ G and n is not prime, methods are discussed to determine whether H(*) is G, H or some intermediate subgroup between H and G.

Perfect colorings of patterns with multiple orbits

“…2(b) and 2(c). Similar figures may be found in the work of Felix & Junio (2015). For both figures, the outer vertices are colored using the same partition P 2 ¼ fJx 2 ; aJx 2 ; a 2 Jx 2 g as in Fig.…”

“…We now proceed to colorings where the orbits share colors. An observation due to Felix & Junio (2015) says that if there is color-sharing between orbits, it is not enough for each orbit to be perfectly colored to make P a perfect coloring.…”

“…However, in the work of Felix & Junio (2015), patterns with orbits that share colors are discussed, and were found to behave quite differently. In particular, it was found that when there is color-sharing between orbits, it was not enough to simply take the intersection of the color groups, because the color permutation induced by a symmetry element on an orbit may be different from the color permutation it induces on another orbit.…”

Symmetry groups of two-way twofold and three-way threefold fabrics

“…Let X be a set of tiles in T . One way to obtain a coloring of T with a color group G � � � G is through the method presented by Felix & Junio (2015), De Las Pen ˜as et al (2006). Using this method one may partition the set of tile orbit representatives ft j g n j¼1 of X under G � as [ s i¼1 X i and obtain the coloring given by the partition…”

Local and global color symmetries of a symmetrical pattern

“…One of the problems in color symmetry is the classification and enumeration of colored symmetrical patterns. Several papers in the literature are dedicated to addressing this problem (see De Las Peñ as et al, 1999a,b, 2006Evidente, 2012;Felix & Junio, 2015;Loeb, 1978;Schwarzenberger, 1984;Senechal, 1979 and references therein). In these studies, colors are added to different elements of the pattern (e.g.…”